1. We are given a $3 \times 3$ grid with integers from 1 to 9, each appearing exactly once. The product of the integers in the first row is 60, and the product of the integers in the second row is 96. We need to find the sum of the integers in the third row.

2. First, calculate the product of all integers from 1 to 9:
$$9! = 1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 = 362880$$

3. The product of the integers in the third row can be found by dividing the total product by the product of the integers in the first and second rows:
$$\text{Product of third row} = \frac{9!}{60 \times 96}$$

4. Calculate $60 \times 96$:
$$60 \times 96 = 5760$$

5. Now, divide $362880$ by $5760$:
$$\frac{362880}{5760} = 63$$

6. Factorize 63 to find the integers in the third row:
$$63 = 3^2 \times 7$$

7. Since each integer in the grid is distinct and must be from 1 to 9, the factors 3, 3, and 7 must correspond to the integers 1, 7, and 9 (since 3 appears twice, it must be 1 and 9).

8. Therefore, the integers in the third row are 1, 7, and 9 in some order.

9. The sum of the integers in the third row is:
$$1 + 7 + 9 = 17$$

Conclusion:
$$\boxed{17}$$